The only way we own a web of our own is to develop much more of a culture of leaving smallish machines online all the time. Imagine something like Tor or BitTorrent, but everyone has a very simple way of running their own node for content hosting. That always-on device? To get critical mass, instead of just the nerds, you'd need it to ship with devices which are always-on, like routers/gateways, smart TV's. Then you'…
This is very right. There are two obstacles. (1) Security. An always-on, externally accessible device will always be a target for breaking in. You want the device to be bulletproof, and to have defense in depth, so that breaking into one service does not affect anything else. Something like Proxmox that works on low-end hardware and is as easy to administer as a mobile phone would do. We are somehow far from this yet…
Every phone/device is their own server, they connect with a web socket to the preferred station which is typically a server online serving as bridge.
There is no need to be always connected to a server, you can also connect locally on the WiFi, BLE or even USB-C cables (discovery is automatic).
From there are internal apps for sharing static websites, chat, blogs, files and so forth.